ab-angle->ABCF B

Percentage Accurate: 54.5% → 66.8%

Time: 53.0s

Alternatives: 8

Speedup: 5.5×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_0\right) \cdot \cos t_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_0\right) \cdot \cos t_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := {\left(\sqrt{\pi}\right)}^{2}\\ t_1 := \left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;{b}^{2} \leq 10^{+53}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left|\cos \left(t_0 \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \cos \left(angle \cdot \left(-0.005555555555555556 \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (pow (sqrt PI) 2.0))
        (t_1
         (* (- a b) (* (+ b a) (sin (* angle (* PI -0.005555555555555556)))))))
   (if (<= (pow b 2.0) 1e+53)
     (* t_1 (* 2.0 (fabs (cos (* t_0 (* angle -0.005555555555555556))))))
     (* t_1 (* 2.0 (cos (* angle (* -0.005555555555555556 t_0))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = pow(sqrt(((double) M_PI)), 2.0);
	double t_1 = (a - b) * ((b + a) * sin((angle * (((double) M_PI) * -0.005555555555555556))));
	double tmp;
	if (pow(b, 2.0) <= 1e+53) {
		tmp = t_1 * (2.0 * fabs(cos((t_0 * (angle * -0.005555555555555556)))));
	} else {
		tmp = t_1 * (2.0 * cos((angle * (-0.005555555555555556 * t_0))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.pow(Math.sqrt(Math.PI), 2.0);
	double t_1 = (a - b) * ((b + a) * Math.sin((angle * (Math.PI * -0.005555555555555556))));
	double tmp;
	if (Math.pow(b, 2.0) <= 1e+53) {
		tmp = t_1 * (2.0 * Math.abs(Math.cos((t_0 * (angle * -0.005555555555555556)))));
	} else {
		tmp = t_1 * (2.0 * Math.cos((angle * (-0.005555555555555556 * t_0))));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = math.pow(math.sqrt(math.pi), 2.0)
	t_1 = (a - b) * ((b + a) * math.sin((angle * (math.pi * -0.005555555555555556))))
	tmp = 0
	if math.pow(b, 2.0) <= 1e+53:
		tmp = t_1 * (2.0 * math.fabs(math.cos((t_0 * (angle * -0.005555555555555556)))))
	else:
		tmp = t_1 * (2.0 * math.cos((angle * (-0.005555555555555556 * t_0))))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = sqrt(pi) ^ 2.0
	t_1 = Float64(Float64(a - b) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * -0.005555555555555556)))))
	tmp = 0.0
	if ((b ^ 2.0) <= 1e+53)
		tmp = Float64(t_1 * Float64(2.0 * abs(cos(Float64(t_0 * Float64(angle * -0.005555555555555556))))));
	else
		tmp = Float64(t_1 * Float64(2.0 * cos(Float64(angle * Float64(-0.005555555555555556 * t_0)))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = sqrt(pi) ^ 2.0;
	t_1 = (a - b) * ((b + a) * sin((angle * (pi * -0.005555555555555556))));
	tmp = 0.0;
	if ((b ^ 2.0) <= 1e+53)
		tmp = t_1 * (2.0 * abs(cos((t_0 * (angle * -0.005555555555555556)))));
	else
		tmp = t_1 * (2.0 * cos((angle * (-0.005555555555555556 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 1e+53], N[(t$95$1 * N[(2.0 * N[Abs[N[Cos[N[(t$95$0 * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Cos[N[(angle * N[(-0.005555555555555556 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 9.9999999999999999e52

   1. Initial program 56.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 57.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 57.7%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 57.2%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 57.2%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 57.2%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 62.6%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 62.6%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 62.6%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 62.6%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 62.2%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 62.2%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 63.1%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 63.1%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)} \cdot \sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)}\right) \]

      2. sqrt-unprod 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}}\right) \]

      3. pow2 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt{\color{blue}{{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{2}}}\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\sqrt{{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{2}}}\right) \]
   11. Step-by-step derivation

       1. unpow2 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt{\color{blue}{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}}\right) \]

       2. rem-sqrt-square 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left|\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right|}\right) \]

       3. associate-*r* 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right|\right) \]

       4. *-commutative 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right) \]

       5. associate-*r* 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right|\right) \]

       6. *-commutative 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)}\right|\right) \]

   12. Simplified 68.6%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|}\right) \]
   13. Step-by-step derivation

       1. add-sqr-sqrt 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|\right) \]

       2. pow2 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|\right) \]

   14. Applied egg-rr 68.6%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|\right) \]

   if 9.9999999999999999e52 < (pow.f64 b 2)

   1. Initial program 39.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 41.5%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 41.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 41.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 56.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 56.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 55.3%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 55.3%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 55.3%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 73.9%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 73.9%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 73.9%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 73.9%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 75.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 75.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 75.1%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 70.3%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|\right) \]

      2. pow2 70.3%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|\right) \]

   10. Applied egg-rr 76.2%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot -0.005555555555555556\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{+53}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left({\left(\sqrt{\pi}\right)}^{2} \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(-0.005555555555555556 \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;{b}^{2} \leq 10^{+53}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \left|\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \cos \left(angle \cdot \left(-0.005555555555555556 \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0
         (* (- a b) (* (+ b a) (sin (* angle (* PI -0.005555555555555556)))))))
   (if (<= (pow b 2.0) 1e+53)
     (* t_0 (* 2.0 (fabs (cos (* PI (* angle -0.005555555555555556))))))
     (*
      t_0
      (* 2.0 (cos (* angle (* -0.005555555555555556 (pow (sqrt PI) 2.0)))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (a - b) * ((b + a) * sin((angle * (((double) M_PI) * -0.005555555555555556))));
	double tmp;
	if (pow(b, 2.0) <= 1e+53) {
		tmp = t_0 * (2.0 * fabs(cos((((double) M_PI) * (angle * -0.005555555555555556)))));
	} else {
		tmp = t_0 * (2.0 * cos((angle * (-0.005555555555555556 * pow(sqrt(((double) M_PI)), 2.0)))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * ((b + a) * Math.sin((angle * (Math.PI * -0.005555555555555556))));
	double tmp;
	if (Math.pow(b, 2.0) <= 1e+53) {
		tmp = t_0 * (2.0 * Math.abs(Math.cos((Math.PI * (angle * -0.005555555555555556)))));
	} else {
		tmp = t_0 * (2.0 * Math.cos((angle * (-0.005555555555555556 * Math.pow(Math.sqrt(Math.PI), 2.0)))));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = (a - b) * ((b + a) * math.sin((angle * (math.pi * -0.005555555555555556))))
	tmp = 0
	if math.pow(b, 2.0) <= 1e+53:
		tmp = t_0 * (2.0 * math.fabs(math.cos((math.pi * (angle * -0.005555555555555556)))))
	else:
		tmp = t_0 * (2.0 * math.cos((angle * (-0.005555555555555556 * math.pow(math.sqrt(math.pi), 2.0)))))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * -0.005555555555555556)))))
	tmp = 0.0
	if ((b ^ 2.0) <= 1e+53)
		tmp = Float64(t_0 * Float64(2.0 * abs(cos(Float64(pi * Float64(angle * -0.005555555555555556))))));
	else
		tmp = Float64(t_0 * Float64(2.0 * cos(Float64(angle * Float64(-0.005555555555555556 * (sqrt(pi) ^ 2.0))))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = (a - b) * ((b + a) * sin((angle * (pi * -0.005555555555555556))));
	tmp = 0.0;
	if ((b ^ 2.0) <= 1e+53)
		tmp = t_0 * (2.0 * abs(cos((pi * (angle * -0.005555555555555556)))));
	else
		tmp = t_0 * (2.0 * cos((angle * (-0.005555555555555556 * (sqrt(pi) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 1e+53], N[(t$95$0 * N[(2.0 * N[Abs[N[Cos[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Cos[N[(angle * N[(-0.005555555555555556 * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 9.9999999999999999e52

   1. Initial program 56.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 57.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 57.7%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 57.2%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 57.2%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 57.2%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 62.6%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 62.6%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 62.6%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 62.6%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 62.2%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 62.2%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 63.1%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 63.1%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)} \cdot \sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)}\right) \]

      2. sqrt-unprod 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}}\right) \]

      3. pow2 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt{\color{blue}{{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{2}}}\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\sqrt{{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{2}}}\right) \]
   11. Step-by-step derivation

       1. unpow2 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt{\color{blue}{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}}\right) \]

       2. rem-sqrt-square 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left|\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right|}\right) \]

       3. associate-*r* 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right|\right) \]

       4. *-commutative 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right) \]

       5. associate-*r* 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right|\right) \]

       6. *-commutative 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)}\right|\right) \]

   12. Simplified 68.6%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|}\right) \]

   if 9.9999999999999999e52 < (pow.f64 b 2)

   1. Initial program 39.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 41.5%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 41.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 41.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 56.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 56.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 55.3%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 55.3%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 55.3%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 73.9%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 73.9%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 73.9%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 73.9%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 75.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 75.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 75.1%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 70.3%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|\right) \]

      2. pow2 70.3%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|\right) \]

   10. Applied egg-rr 76.2%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{2}} \cdot -0.005555555555555556\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{+53}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(-0.005555555555555556 \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 3: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot -0.005555555555555556\right)\\ t_1 := \left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin t_0\right)\\ \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+247}:\\ \;\;\;\;2 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \cos t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI -0.005555555555555556)))
        (t_1 (* (- a b) (* (+ b a) (sin t_0)))))
   (if (<= (- (pow b 2.0) (pow a 2.0)) -2e+247)
     (* 2.0 t_1)
     (* t_1 (* 2.0 (cos t_0))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * -0.005555555555555556);
	double t_1 = (a - b) * ((b + a) * sin(t_0));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -2e+247) {
		tmp = 2.0 * t_1;
	} else {
		tmp = t_1 * (2.0 * cos(t_0));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * -0.005555555555555556);
	double t_1 = (a - b) * ((b + a) * Math.sin(t_0));
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -2e+247) {
		tmp = 2.0 * t_1;
	} else {
		tmp = t_1 * (2.0 * Math.cos(t_0));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = angle * (math.pi * -0.005555555555555556)
	t_1 = (a - b) * ((b + a) * math.sin(t_0))
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -2e+247:
		tmp = 2.0 * t_1
	else:
		tmp = t_1 * (2.0 * math.cos(t_0))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * -0.005555555555555556))
	t_1 = Float64(Float64(a - b) * Float64(Float64(b + a) * sin(t_0)))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -2e+247)
		tmp = Float64(2.0 * t_1);
	else
		tmp = Float64(t_1 * Float64(2.0 * cos(t_0)));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * -0.005555555555555556);
	t_1 = (a - b) * ((b + a) * sin(t_0));
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -2e+247)
		tmp = 2.0 * t_1;
	else
		tmp = t_1 * (2.0 * cos(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -2e+247], N[(2.0 * t$95$1), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -1.9999999999999999e247

   1. Initial program 43.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 46.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 46.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 46.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 46.0%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 43.5%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 43.5%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 43.5%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 66.3%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 66.3%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 66.3%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 66.3%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 67.4%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 67.4%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]

   9. Taylor expanded in angle around 0 79.7%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{1}\right) \]

   if -1.9999999999999999e247 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 50.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 51.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 51.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 51.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 60.6%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 60.6%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 60.3%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 60.3%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 60.3%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 68.4%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 68.4%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 68.4%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 68.4%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 68.9%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 68.9%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+247}:\\ \;\;\;\;2 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 4: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+247}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0
         (* (- a b) (* (+ b a) (sin (* angle (* PI -0.005555555555555556)))))))
   (if (<= (- (pow b 2.0) (pow a 2.0)) -2e+247)
     (* 2.0 t_0)
     (* t_0 (* 2.0 (cos (* PI (* angle -0.005555555555555556))))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = (a - b) * ((b + a) * sin((angle * (((double) M_PI) * -0.005555555555555556))));
	double tmp;
	if ((pow(b, 2.0) - pow(a, 2.0)) <= -2e+247) {
		tmp = 2.0 * t_0;
	} else {
		tmp = t_0 * (2.0 * cos((((double) M_PI) * (angle * -0.005555555555555556))));
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = (a - b) * ((b + a) * Math.sin((angle * (Math.PI * -0.005555555555555556))));
	double tmp;
	if ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) <= -2e+247) {
		tmp = 2.0 * t_0;
	} else {
		tmp = t_0 * (2.0 * Math.cos((Math.PI * (angle * -0.005555555555555556))));
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = (a - b) * ((b + a) * math.sin((angle * (math.pi * -0.005555555555555556))))
	tmp = 0
	if (math.pow(b, 2.0) - math.pow(a, 2.0)) <= -2e+247:
		tmp = 2.0 * t_0
	else:
		tmp = t_0 * (2.0 * math.cos((math.pi * (angle * -0.005555555555555556))))
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = Float64(Float64(a - b) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * -0.005555555555555556)))))
	tmp = 0.0
	if (Float64((b ^ 2.0) - (a ^ 2.0)) <= -2e+247)
		tmp = Float64(2.0 * t_0);
	else
		tmp = Float64(t_0 * Float64(2.0 * cos(Float64(pi * Float64(angle * -0.005555555555555556)))));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = (a - b) * ((b + a) * sin((angle * (pi * -0.005555555555555556))));
	tmp = 0.0;
	if (((b ^ 2.0) - (a ^ 2.0)) <= -2e+247)
		tmp = 2.0 * t_0;
	else
		tmp = t_0 * (2.0 * cos((pi * (angle * -0.005555555555555556))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision], -2e+247], N[(2.0 * t$95$0), $MachinePrecision], N[(t$95$0 * N[(2.0 * N[Cos[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -1.9999999999999999e247

   1. Initial program 43.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 46.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 46.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 46.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 46.0%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 46.0%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 43.5%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 43.5%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 43.5%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 66.3%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 66.3%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 66.3%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 66.3%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 67.4%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 67.4%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]

   9. Taylor expanded in angle around 0 79.7%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{1}\right) \]

   if -1.9999999999999999e247 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 50.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 51.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 51.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 51.4%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 60.6%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 60.6%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 60.3%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 60.3%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 60.3%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 68.4%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 68.4%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 68.4%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 68.4%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 68.9%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 68.9%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 68.7%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]

   9. Taylor expanded in angle around inf 68.9%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* 68.9%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right) \]

       2. *-commutative 68.9%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)}\right) \]

   11. Simplified 68.9%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\cos \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{+247}:\\ \;\;\;\;2 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 5: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\\ t_1 := \left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;{b}^{2} \leq 10^{+53}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left|t_0\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* PI (* angle -0.005555555555555556))))
        (t_1
         (* (- a b) (* (+ b a) (sin (* angle (* PI -0.005555555555555556)))))))
   (if (<= (pow b 2.0) 1e+53) (* t_1 (* 2.0 (fabs t_0))) (* t_1 (* 2.0 t_0)))))

C

b = abs(b);
double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) * (angle * -0.005555555555555556)));
	double t_1 = (a - b) * ((b + a) * sin((angle * (((double) M_PI) * -0.005555555555555556))));
	double tmp;
	if (pow(b, 2.0) <= 1e+53) {
		tmp = t_1 * (2.0 * fabs(t_0));
	} else {
		tmp = t_1 * (2.0 * t_0);
	}
	return tmp;
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	double t_0 = Math.cos((Math.PI * (angle * -0.005555555555555556)));
	double t_1 = (a - b) * ((b + a) * Math.sin((angle * (Math.PI * -0.005555555555555556))));
	double tmp;
	if (Math.pow(b, 2.0) <= 1e+53) {
		tmp = t_1 * (2.0 * Math.abs(t_0));
	} else {
		tmp = t_1 * (2.0 * t_0);
	}
	return tmp;
}

Python

b = abs(b)
def code(a, b, angle):
	t_0 = math.cos((math.pi * (angle * -0.005555555555555556)))
	t_1 = (a - b) * ((b + a) * math.sin((angle * (math.pi * -0.005555555555555556))))
	tmp = 0
	if math.pow(b, 2.0) <= 1e+53:
		tmp = t_1 * (2.0 * math.fabs(t_0))
	else:
		tmp = t_1 * (2.0 * t_0)
	return tmp

Julia

b = abs(b)
function code(a, b, angle)
	t_0 = cos(Float64(pi * Float64(angle * -0.005555555555555556)))
	t_1 = Float64(Float64(a - b) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * -0.005555555555555556)))))
	tmp = 0.0
	if ((b ^ 2.0) <= 1e+53)
		tmp = Float64(t_1 * Float64(2.0 * abs(t_0)));
	else
		tmp = Float64(t_1 * Float64(2.0 * t_0));
	end
	return tmp
end

MATLAB

b = abs(b)
function tmp_2 = code(a, b, angle)
	t_0 = cos((pi * (angle * -0.005555555555555556)));
	t_1 = (a - b) * ((b + a) * sin((angle * (pi * -0.005555555555555556))));
	tmp = 0.0;
	if ((b ^ 2.0) <= 1e+53)
		tmp = t_1 * (2.0 * abs(t_0));
	else
		tmp = t_1 * (2.0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi * N[(angle * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 1e+53], N[(t$95$1 * N[(2.0 * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b 2) < 9.9999999999999999e52

   1. Initial program 56.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 57.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 57.7%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 57.7%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 57.2%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 57.2%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 57.2%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 62.6%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 62.6%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 62.6%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 62.6%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 62.2%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 62.2%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 63.1%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 63.1%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 57.0%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left(\sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)} \cdot \sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)}\right) \]

      2. sqrt-unprod 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\sqrt{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}}\right) \]

      3. pow2 68.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt{\color{blue}{{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{2}}}\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\sqrt{{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}^{2}}}\right) \]
   11. Step-by-step derivation

       1. unpow2 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \sqrt{\color{blue}{\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right) \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}}\right) \]

       2. rem-sqrt-square 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left|\cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right|}\right) \]

       3. associate-*r* 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right|\right) \]

       4. *-commutative 68.7%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right|\right) \]

       5. associate-*r* 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right|\right) \]

       6. *-commutative 68.6%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \color{blue}{\left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)}\right|\right) \]

   12. Simplified 68.6%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left|\cos \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right|}\right) \]

   if 9.9999999999999999e52 < (pow.f64 b 2)

   1. Initial program 39.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Simplified 41.5%

      \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 41.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

      2. unpow2 41.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

      3. difference-of-squares 56.5%

         \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   5. Applied egg-rr 56.5%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

   6. Taylor expanded in angle around inf 55.3%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 55.3%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

      2. *-commutative 55.3%

         \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

      3. associate-*r* 73.9%

         \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      4. *-commutative 73.9%

         \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      5. *-commutative 73.9%

         \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      6. *-commutative 73.9%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      7. associate-*r* 75.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

      8. *-commutative 75.7%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

      9. associate-*r* 75.1%

         \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

   8. Simplified 75.1%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]

   9. Taylor expanded in angle around inf 75.7%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r* 75.5%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right)}\right) \]

       2. *-commutative 75.5%

          \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)}\right) \]

   11. Simplified 75.5%

       \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\cos \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{+53}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \left|\cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(\pi \cdot \left(angle \cdot -0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 6: 56.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* (* 2.0 (sin (* angle (/ PI -180.0)))) (* (- a b) (+ b a))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return (2.0 * sin((angle * (((double) M_PI) / -180.0)))) * ((a - b) * (b + a));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return (2.0 * Math.sin((angle * (Math.PI / -180.0)))) * ((a - b) * (b + a));
}

Python

b = abs(b)
def code(a, b, angle):
	return (2.0 * math.sin((angle * (math.pi / -180.0)))) * ((a - b) * (b + a))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(Float64(2.0 * sin(Float64(angle * Float64(pi / -180.0)))) * Float64(Float64(a - b) * Float64(b + a)))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = (2.0 * sin((angle * (pi / -180.0)))) * ((a - b) * (b + a));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(N[(2.0 * N[Sin[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 50.1%

   \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 50.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

   2. unpow2 50.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

   3. difference-of-squares 57.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

5. Applied egg-rr 57.1%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

6. Taylor expanded in angle around 0 57.2%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\color{blue}{1} \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \]

7. Final simplification 57.2%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\left(a - b\right) \cdot \left(b + a\right)\right) \]

Alternative 7: 65.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ 2 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* 2.0 (* (- a b) (* (+ b a) (sin (* angle (* PI -0.005555555555555556)))))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return 2.0 * ((a - b) * ((b + a) * sin((angle * (((double) M_PI) * -0.005555555555555556)))));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return 2.0 * ((a - b) * ((b + a) * Math.sin((angle * (Math.PI * -0.005555555555555556)))));
}

Python

b = abs(b)
def code(a, b, angle):
	return 2.0 * ((a - b) * ((b + a) * math.sin((angle * (math.pi * -0.005555555555555556)))))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(a - b) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * -0.005555555555555556))))))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = 2.0 * ((a - b) * ((b + a) * sin((angle * (pi * -0.005555555555555556)))));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(2.0 * N[(N[(a - b), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 50.1%

   \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 50.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

   2. unpow2 50.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

   3. difference-of-squares 57.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

5. Applied egg-rr 57.1%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

6. Taylor expanded in angle around inf 56.3%

   \[\leadsto \color{blue}{2 \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 56.3%

      \[\leadsto \color{blue}{\left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

   2. *-commutative 56.3%

      \[\leadsto \color{blue}{\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   3. associate-*r* 67.9%

      \[\leadsto \color{blue}{\left(\left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right) \cdot \left(a - b\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   4. *-commutative 67.9%

      \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a + b\right)\right)\right)} \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   5. *-commutative 67.9%

      \[\leadsto \left(\left(a - b\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \sin \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. *-commutative 67.9%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   7. associate-*r* 68.5%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right)\right) \cdot \left(2 \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   8. *-commutative 68.5%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right)}\right) \]

   9. associate-*r* 68.7%

      \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)}\right) \]

8. Simplified 68.7%

   \[\leadsto \color{blue}{\left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \cos \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)} \]

9. Taylor expanded in angle around 0 68.8%

   \[\leadsto \left(\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \cdot \left(2 \cdot \color{blue}{1}\right) \]

10. Final simplification 68.8%

    \[\leadsto 2 \cdot \left(\left(a - b\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right) \]

Alternative 8: 54.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} b = |b|\\ \\ -0.011111111111111112 \cdot \left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \pi\right)\right) \end{array} \]

FPCore

NOTE: b should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (* -0.011111111111111112 (* (* (- a b) (+ b a)) (* angle PI))))

C

b = abs(b);
double code(double a, double b, double angle) {
	return -0.011111111111111112 * (((a - b) * (b + a)) * (angle * ((double) M_PI)));
}

Java

b = Math.abs(b);
public static double code(double a, double b, double angle) {
	return -0.011111111111111112 * (((a - b) * (b + a)) * (angle * Math.PI));
}

Python

b = abs(b)
def code(a, b, angle):
	return -0.011111111111111112 * (((a - b) * (b + a)) * (angle * math.pi))

Julia

b = abs(b)
function code(a, b, angle)
	return Float64(-0.011111111111111112 * Float64(Float64(Float64(a - b) * Float64(b + a)) * Float64(angle * pi)))
end

MATLAB

b = abs(b)
function tmp = code(a, b, angle)
	tmp = -0.011111111111111112 * (((a - b) * (b + a)) * (angle * pi));
end

Wolfram

NOTE: b should be positive before calling this function
code[a_, b_, angle_] := N[(-0.011111111111111112 * N[(N[(N[(a - b), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Simplified 50.1%

   \[\leadsto \color{blue}{\left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left({a}^{2} - {b}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 50.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(\color{blue}{a \cdot a} - {b}^{2}\right)\right) \]

   2. unpow2 50.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \left(a \cdot a - \color{blue}{b \cdot b}\right)\right) \]

   3. difference-of-squares 57.1%

      \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

5. Applied egg-rr 57.1%

   \[\leadsto \left(2 \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right) \cdot \left(\cos \left(angle \cdot \frac{\pi}{-180}\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)}\right) \]

6. Taylor expanded in angle around 0 52.1%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 52.1%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(a - b\right)\right)\right)} \]

   2. *-commutative 52.1%

      \[\leadsto -0.011111111111111112 \cdot \color{blue}{\left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \left(angle \cdot \pi\right)\right)} \]

8. Simplified 52.1%

   \[\leadsto \color{blue}{-0.011111111111111112 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \left(angle \cdot \pi\right)\right)} \]

9. Final simplification 52.1%

   \[\leadsto -0.011111111111111112 \cdot \left(\left(\left(a - b\right) \cdot \left(b + a\right)\right) \cdot \left(angle \cdot \pi\right)\right) \]

Reproduce

herbie shell --seed 2023333 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))